Question 734187: Find a third degree polynomial that has x=1 as its only real zero and this zero has a: multiplicity one (1) b: multiplicity three (3)
Answer by Positive_EV(69)   (Show Source):
You can put this solution on YOUR website! All third degree polynomials have three roots. If x=1 is a root, the polynomial can be factored so that (x-1) is a factor.
If it is the only real root, and it's multiplicity one, that means that the other two roots are imaginary, and are also conjugates. The simplest pair of imaginary conjugate roots are i and -i. The quadratic equation with roots i and -i is (x+i)(x-i) = 0, or x^2+1 = 0.
The third degree polynomial with 1, i and -i as roots is (x-1)(x^2+1) = 0, or x^3-x^2+x-1 = 0. There are other solutions, depending on what imaginary roots you use.
If 1 has multiplicity three, that means that it is a triple root. The one and only one cubic equation with a root of 1 that has multiplicity three is (x-1)(x-1)(x-1) = 0, or x^3-3x^2+3x-1.
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